International Mathematics Competition
for University Students
2025

IMC2025: Day 2, Problem 8

Problem 8. For an \(\displaystyle n\times n\) real matrix \(\displaystyle A\in M_n(\RR)\), denote by \(\displaystyle A^{\mathsf{R}}\) its counter-clockwise \(\displaystyle 90^\circ\) rotation. For example,

\(\displaystyle \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}^{\mathsf{R}} = \begin{bmatrix} 3 & 6 & 9 \\ 2 & 5 & 8 \\ 1 & 4 & 7 \end{bmatrix}. \)

Prove that if \(\displaystyle A = A^{\mathsf{R}}\) then for any eigenvalue \(\displaystyle \lambda\) of \(\displaystyle A\), we have \(\displaystyle \operatorname{Re}\lambda = 0\) or \(\displaystyle \operatorname{Im}\lambda = 0\).

Jan Kuś, University of Warwick

  Solution